| L(s) = 1 | − 9·3-s − 83.5i·5-s − 187. i·7-s + 81·9-s + 786. i·11-s + (259. − 551. i)13-s + 751. i·15-s + 923.·17-s + 434. i·19-s + 1.68e3i·21-s − 3.79e3·23-s − 3.85e3·25-s − 729·27-s − 3.73e3·29-s + 2.69e3i·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.49i·5-s − 1.44i·7-s + 0.333·9-s + 1.95i·11-s + (0.426 − 0.904i)13-s + 0.862i·15-s + 0.775·17-s + 0.276i·19-s + 0.834i·21-s − 1.49·23-s − 1.23·25-s − 0.192·27-s − 0.824·29-s + 0.502i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.426 - 0.904i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.426 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.5686126303\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5686126303\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 9T \) |
| 13 | \( 1 + (-259. + 551. i)T \) |
| good | 5 | \( 1 + 83.5iT - 3.12e3T^{2} \) |
| 7 | \( 1 + 187. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 786. iT - 1.61e5T^{2} \) |
| 17 | \( 1 - 923.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 434. iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 3.79e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.73e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.69e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + 1.73e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.21e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 1.80e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 5.64e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 3.05e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.89e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 9.01e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.81e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 8.03e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + 5.86e3iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 1.72e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.47e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 4.57e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 7.79e4iT - 8.58e9T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.01733038620624437497611249574, −9.364807326295470978875590298694, −7.903098977010899854913347713937, −7.61644291505357854844930632169, −6.35633110555998602886420698710, −5.22403859408740727997270386529, −4.53393065547026563146123251789, −3.77461463984287523264616705416, −1.69403688479111205930761129494, −0.963380988637288398884528926225,
0.15289587078868632381550844744, 1.90535677807436582934820865345, 2.97411696913374806669184589846, 3.80999490007897388043286719653, 5.57629880551796990781445779692, 5.98426684581351002022197087545, 6.72310260171302097910626339549, 7.943395724716053331477889296917, 8.799799970145680370012061151907, 9.776404753296376811803481659705
